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 A084231 Numbers k such that the root-mean-square value of 1, 2, ..., k, i.e., sqrt((1/k)*Sum_{j=1..k} j^2), is an integer. 6
 1, 337, 65521, 12710881, 2465845537, 478361323441, 92799630902161, 18002650033695937, 3492421306906109761, 677511730889751597841, 131433783371304903871537, 25497476462302261599480481, 4946378999903267445395341921, 959572028504771582145096852337 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Equivalently, numbers k such that sqrt((k+1)*(2*k+1)/6) is an integer. LINKS Indranil Ghosh, Table of n, a(n) for n = 1..437 Peter Khoury and Gerard D. Koffi, Continued fractions and their application to solving Pell’s equations (2009) Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (195,-195,1). FORMULA a(n) = ((7/2 + 2*sqrt(3))*(97 + 56*sqrt(3))^n + (7/2 - 2*sqrt(3))*(97 - 56*sqrt(3))^n - 3)/4. a(n) = (floor((7/2 + 2*sqrt(3))*(97 + 56*sqrt(3))^n) - 2)/4. a(n+3) = 195*(a(n+2) - a(n+1)) + a(n). G.f.: x*(1+142*x+x^2)/((1-x)*(1-194*x+x^2)). a(n) = ((7 - 4*sqrt(3))^(1+2n) + (7 + 4*sqrt(3))^(1+2n) - 6)/8. - Peter Pein (peter.pein(AT)dordos.de), Mar 03 2005 a(n) = 195*a(n-1) - 195*a(n-2) + a(n-3), with a(0)=0, a(1)=1, a(2)=337, a(3)=65521. - Harvey P. Dale, Jul 14 2011 EXAMPLE 337 is in the sequence because sqrt((1/337)*Sum_{k=1..337} k^2) is an integer (195=A084232(1)). MATHEMATICA a[n_]:=Expand[((7-4 Sqrt[3])^(1+2n)+(7+4 Sqrt[3])^(1+2n)-6)/8] (* Peter Pein *) CoefficientList[Series[x (1+142x+x^2)/((1-x)(1-194x+x^2)), {x, 0, 30}], x] (* or *) Join[{0}, LinearRecurrence[{195, -195, 1}, {1, 337, 65521}, 30]] (* Harvey P. Dale, Jul 14 2011 *) CROSSREFS Cf. A084232. Sequence in context: A263865 A184202 A194478 * A243483 A234625 A226539 Adjacent sequences: A084228 A084229 A084230 * A084232 A084233 A084234 KEYWORD nonn,easy AUTHOR Ignacio Larrosa Cañestro, May 20 2003 EXTENSIONS One more term from Peter Pein (peter.pein(AT)dordos.de), Mar 03 2005 STATUS approved

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Last modified June 14 21:35 EDT 2024. Contains 373401 sequences. (Running on oeis4.)